Proof of Sequences: Orders and Representations

In summary, the conversation discusses how to prove that the nth term of a sequence of order 2 (or any order k) can be represented as a polynomial. The key is to focus on the highest order term and use the binomial expansion to find the differences between terms.
  • #1
courtrigrad
1,236
2
Hello all

Let us say we are given a sequence of order 2. By order 2 I mean that we have a sequence in which the differences between the terms forms a sequence of order 1, which has a constant difference between terms. How can I prove that the nth term of a sequence of order 2 can be represented as:

an^2 + bn + c?


Or more generally how would I prove that that the nth term of a sequence of order k can be represented as:

an^k + bn^k-1 +... + pn + q?

Any help would we greatly appreciated.

Thanks
 
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  • #2
I think you only need to focus on the highest order term. Just look at the difference [itex]P(n+1) - P(n)[/itex] where P(n) is your sequence and use the first term in the binomial expansion of [itex]P(n+1)[/itex].
 
  • #3
for bringing up this interesting topic! To prove the representation of the nth term of a sequence of order k, we can use mathematical induction. First, we need to show that the representation holds for the initial terms of the sequence. In your example of order 2, we can show that the first two terms (n=1 and n=2) can be represented as a+b+c and a+2b+c, respectively.

Next, we assume that the representation holds for some arbitrary term k, meaning that the kth term can be represented as ak^2 + bk + c. Now, we need to show that the representation also holds for the (k+1)th term. We can use the given order of the sequence to show that the difference between the (k+1)th term and the kth term is a constant. This means that the difference between the (k+1)th term and the kth term can be represented as d, where d is a constant.

Using this information, we can write the (k+1)th term as ak^2 + bk + c + d. Simplifying this expression, we get a(k+1)^2 + b(k+1) + c, which is the desired representation. This completes the proof by induction, showing that the representation holds for all terms of the sequence.

We can extend this proof to sequences of order k by using a similar approach. We would need to show that the representation holds for the first k terms of the sequence, and then use induction to show that it also holds for the (k+1)th term.

I hope this helps and provides some insight into proving the representation of sequences of different orders. Keep exploring and asking questions - that is the key to understanding and mastering mathematics. Good luck!
 

1. What is a sequence in mathematics?

A sequence in mathematics is a list of numbers or objects that follow a specific pattern or rule. Each individual number or object in the sequence is called a term, and the position of the term in the sequence is called its index. Sequences can be finite (with a limited number of terms) or infinite (with an endless number of terms).

2. What is the order of a sequence?

The order of a sequence refers to the specific pattern or rule that determines the values of the terms in the sequence. It can also be thought of as the number of terms in the sequence. For example, the sequence 2, 4, 6, 8, 10 has an order of 5 because it has 5 terms and each term increases by 2.

3. How are sequences represented in mathematics?

Sequences can be represented in different ways, including using mathematical notation, tables, graphs, or recursive formulas. In mathematical notation, sequences are often represented using subscripts to indicate the position or index of the term. For example, the sequence 1, 4, 7, 10 can be represented as a1 = 1, a2 = 4, a3 = 7, a4 = 10.

4. What is the difference between arithmetic and geometric sequences?

Arithmetic and geometric sequences are two common types of sequences. In an arithmetic sequence, each term is calculated by adding a constant value to the previous term. For example, the sequence 3, 6, 9, 12 is an arithmetic sequence with a common difference of 3. In a geometric sequence, each term is calculated by multiplying a constant value to the previous term. For example, the sequence 2, 6, 18, 54 is a geometric sequence with a common ratio of 3.

5. How are sequences used in real-life situations?

Sequences are used in various fields such as mathematics, science, and finance to model and solve real-life problems. For example, in mathematics, sequences are used to find the next term in a pattern or to calculate the sum of a series. In science, sequences are used to model natural phenomena, such as the Fibonacci sequence in biology to describe the growth of a population. In finance, sequences are used to model the growth of investments or to predict future trends in the stock market.

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